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On Mixed Error Estimates for Elliptic Obstacle Problems

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Abstract

We establish in this paper sharp error estimates of residual type for finite element approximation to elliptic obstacle problems. The estimates are of mixed nature, which are neither of a pure a priori form nor of a pure a posteriori form but instead they are combined by an a priori part and an a posteriori part. The key ingredient in our derivation for the mixed error estimates is the use of a new interpolator which enables us to eliminate inactive data from the error estimators. One application of our mixed error estimates is to construct a posteriori error indicators reliable and efficient up to higher order terms, and these indicators are useful in mesh-refinements and adaptive grid generations. In particular, by approximating the a priori part with some a posteriori quantities we can successfully track the free boundary for elliptic obstacle problems.

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Authors and Affiliations

  1. Department of Mathematics, Xiang Tan University, Hunan Province, P.R. China

    Wenbin Liu

  2. CBS and Institute of Mathematics and Statistics, The University of Kent, Canterbury, CT2 7NF, England

    Wenbin Liu

  3. Department of Mathematics, Shanghai University, Shanghai, 200436, P.R.China

    Heping Ma

  4. Department of Mathematics, The Hong Kong Baptist University, Kowloon Tong, Hong Kong

    Tao Tang

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  1. Wenbin Liu
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  2. Heping Ma
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  3. Tao Tang
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Liu, W., Ma, H. & Tang, T. On Mixed Error Estimates for Elliptic Obstacle Problems. Advances in Computational Mathematics 15, 261–283 (2001). https://doi.org/10.1023/A:1014261013164

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